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Theorem spw 2042
Description: Weak version of the specialization scheme sp 2183. Lemma 9 of [KalishMontague] p. 87. While it appears that sp 2183 in its general form does not follow from Tarski's FOL axiom schemes, from this theorem we can prove any instance of sp 2183 having mutually distinct setvar variables and no wff metavariables (see ax12wdemo 2140 for an example of the procedure to eliminate the hypothesis). Other approximations of sp 2183 are spfw 2041 (minimal distinct variable requirements), spnfw 1985 (when 𝑥 is not free in ¬ 𝜑), spvw 1986 (when 𝑥 does not appear in 𝜑), sptruw 1808 (when 𝜑 is true), and spfalw 2005 (when 𝜑 is false). (Contributed by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 27-Feb-2018.)
Hypothesis
Ref Expression
spw.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
spw (∀𝑥𝜑𝜑)
Distinct variable groups:   𝑥,𝑦   𝜓,𝑥   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem spw
StepHypRef Expression
1 ax-5 1912 . 2 𝜓 → ∀𝑥 ¬ 𝜓)
2 ax-5 1912 . 2 (∀𝑥𝜑 → ∀𝑦𝑥𝜑)
3 ax-5 1912 . 2 𝜑 → ∀𝑦 ¬ 𝜑)
4 spw.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
51, 2, 3, 4spfw 2041 1 (∀𝑥𝜑𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wal 1536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782
This theorem is referenced by:  hba1w  2055  spaev  2058  ax12w  2138  bj-ssblem1  33994  bj-ax12w  34017
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